Multilevel measurement models (MMM), an application of hierarchical generalized linear models (HGLM), model the relationship between ability levels estimates and item difficulty parameters, based on examinee responses to items. A benefit of using MMM is the ability to include additional levels in the model to represent a nested data structure, which is common in educational contexts, by using the multilevel framework. Previous research has demonstrated the ability of the one-parameter MMM to accurately recover both item difficulty parameters and examinee ability levels, when using both 2- and 3-level models, under various sample size and test length conditions (Kamata, 1999; Brune, 2011). Parameter invariance of measurement models, that parameter estimates are equivalent regardless of the distribution of the ability levels, is important when the typical assumption of a normal distribution of ability levels in the population may not be correct. An assumption of MMM is that the distribution of examinee abilities, which is represented by the level-2 residuals in the HGLM, is normal. If the distribution of abilities in the population are not normal, as suggested by Micceri (1989), this assumption of MMM is violated, which has been shown to affect the estimation of the level-2 residuals. The current study investigated the parameter invariance of the 2-level 1P-MMM, by examining the accuracy of item difficulty parameter estimates and examinee ability level estimates. Study conditions included the standard normal distribution, as a baseline, and three non-normal distributions having various degrees of skew, in addition to various test lengths and sample sizes, to simulate various testing conditions. The study's results provide evidence for overall parameter invariance of the 2-level 1P-MMM, when accounting for scale indeterminacy from the estimation process, for the study conditions included. Although, the error in the item difficulty parameter and examinee ability level estimates in the study were not of practical importance, there was some evidence that ability distributions may affect the accuracy of parameter estimates for items with difficulties greater than represented in this study. Also, the accuracy of abilities estimates for non-normal distributions seemed less for conditions with greater test lengths and sample sizes, indicating possible increased difficulty in estimating abilities from non-normal distributions. / text

This dissertation has multiple aims in studying hierarchical linear models in biomedical data analysis. In Chapter 1, the novel idea of studying the durations of adolescent growth spurts as a predictor of adulthood obesity is defined, established, and illustrated. The concept of contextual effects modeling is introduced in this first section as we study secular trend of adulthood obesity and how this trend is mitigated by the durations of individual adolescent growth spurts and the secular average length of adolescent growth spurts. It is found that individuals with longer periods of fast height growth in adolescence are more prone to having favorable BMI profiles in adulthood. In Chapter 2 we study the estimation of contextual effects in a hierarchical generalized linear model (HGLM). We simulate data and study the effects using the higher level group sample mean as the estimate for the true mean versus using an Empirical Bayes (EB) approach (Shin and Raudenbush 2010). We study this comparison for logistic, probit, log-linear, ordinal and nominal regression models. We find that in general the EB estimate lends a parameter estimate much closer to the true value, except for cases with very small variability in the upper level, where it is a more complicated situation and there is likely no need for contextual effects analysis. In Chapter 3 the HGLM studies are made clearer with large-scale simulations. These large scale simulations are shown for logistic regression and probit regression models for binary outcome data. With repetition we are able to establish coverage percentages of the confidence intervals of the true contextual effect. Coverage percentages show the percentage of simulations that have confidence intervals containing the true parameter values. Results confirm observations from the preliminary simulations in the previous section of this paper, and an accompanying example of adulthood hypertension shows how these results can be used in an application.

Animal traits differ not only in mean, but also in variation around the mean. For instance, one sire’s daughter group may be very homogeneous, while another sire’s daughters are much more heterogeneous in performance. The difference in residual variance can partially be explained by genetic differences. Models for such genetic heterogeneity of environmental variance include genetic effects for the mean and residual variance, and a correlation between the genetic effects for the mean and residual variance to measure how the residual variance might vary with the mean. The aim of this thesis was to develop a method based on double hierarchical generalized linear models for estimating genetic heteroscedasticity, and to apply it on four traits in two domestic animal species; teat count and litter size in pigs, and milk production and somatic cell count in dairy cows. The method developed is fast and has been implemented in software that is widely used in animal breeding, which makes it convenient to use. It is based on an approximation of double hierarchical generalized linear models by normal distributions. When having repeated observations on individuals or genetic groups, the estimates were found to be unbiased. For the traits studied, the estimated heritability values for the mean and the residual variance, and the genetic coefficients of variation, were found in the usual ranges reported. The genetic correlation between mean and residual variance was estimated for the pig traits only, and was found to be favorable for litter size, but unfavorable for teat count.

Dentre as possíveis propostas encontradas na literatura estatística para analisar dados oriundos de estudos com observações correlacionadas, estão os modelos condicionais e os modelos marginais. Diversas técnicas têm sido propostas para a análise de diagnóstico nesses modelos. O objetivo deste trabalho é apresentar algumas das técnicas de diagnóstico disponíveis para os dois tipos de modelos e avaliá-las por meio de estudos de simulação. As técnicas apresentadas também foram aplicadas em um conjunto de dados reais. / Conditional and marginal models are among the possibilities in statistical literature to analyze data from studies with correlated observations. Several techniques have been proposed for diagnostic analysis in these models. The objective of this work is to present some of the diagnostic techniques available for both modeling approaches and to evaluate them by simulation studies. The presented techniques were also applied in a real dataset.

Dentre as possíveis propostas encontradas na literatura estatística para analisar dados oriundos de estudos com observações correlacionadas, estão os modelos condicionais e os modelos marginais. Diversas técnicas têm sido propostas para a análise de diagnóstico nesses modelos. O objetivo deste trabalho é apresentar algumas das técnicas de diagnóstico disponíveis para os dois tipos de modelos e avaliá-las por meio de estudos de simulação. As técnicas apresentadas também foram aplicadas em um conjunto de dados reais. / Conditional and marginal models are among the possibilities in statistical literature to analyze data from studies with correlated observations. Several techniques have been proposed for diagnostic analysis in these models. The objective of this work is to present some of the diagnostic techniques available for both modeling approaches and to evaluate them by simulation studies. The presented techniques were also applied in a real dataset.